Question

your turn!
practice: multiply rational expressions
which of the following is equivalent to $2x^3 \cdot 3x^5$?
choose 1 answer:
a $5x^8$
b $6x^8$
c $6x^{15}$
d $8x^{15}$
check
explain

practice: change bases
if $a^{\frac{b}{2}} = 25$ for positive integers $a$ and $b$, what is one possible value of $b$?
check
explain

Explanation:

First Sub - Question (Multiply Rational Expressions)

Step 1: Multiply the coefficients

For the expression \(2x^{3}\cdot3x^{5}\), first multiply the coefficients 2 and 3. So, \(2\times3 = 6\).

Step 2: Multiply the variables using exponent rule

When multiplying variables with exponents, we use the rule \(x^{m}\cdot x^{n}=x^{m + n}\). Here, \(x^{3}\cdot x^{5}=x^{3+5}=x^{8}\).

Step 3: Combine the results

Combine the coefficient and the variable part. We get \(6x^{8}\).

We know that \(a^{\frac{b}{2}} = 25\). We can rewrite 25 as a perfect square. Since \(5^{2}=25\) and \(25^{1}=25\), let's consider the form of the exponent.
If we rewrite \(a^{\frac{b}{2}}\) as \((a^{\frac{1}{2}})^{b}\) or \((\sqrt{a})^{b}\). Also, we can think of 25 as \(5^{2}\) or \(25^{1}\).
Case 1: Let \(a = 25\), then \(a^{\frac{b}{2}}=25^{\frac{b}{2}}\). We want \(25^{\frac{b}{2}} = 25^{1}\), so \(\frac{b}{2}=1\), then \(b = 2\).
Case 2: Let \(a = 5\), then \(a^{\frac{b}{2}}=5^{\frac{b}{2}}\). We want \(5^{\frac{b}{2}}=25 = 5^{2}\), so \(\frac{b}{2}=2\), then \(b = 4\).
(We can also have other cases, but we will take one possible value, for example, \(b = 2\) or \(b = 4\) etc.)

Answer:

B. \(6x^{8}\)

Second Sub - Question (Change Bases)